Question

1. given: ∠s ≅ ∠v, t is the midpoint of \\(\overline{sv}\\). prove: \\(\triangle rts \cong \triangle utv\\)

Explanation:

Step1: State given midpoint property

Since $T$ is the midpoint of $\overline{SV}$, $\overline{ST} \cong \overline{VT}$.

Step2: Identify vertical angles

$\angle RTS$ and $\angle UTV$ are vertical angles, so $\angle RTS \cong \angle UTV$.

Step3: List all congruent parts

We have:

1. $\angle S \cong \angle V$ (Given)
2. $\overline{ST} \cong \overline{VT}$ (From Step1)
3. $\angle RTS \cong \angle UTV$ (From Step2)

Step4: Apply ASA congruence

By the Angle-Side-Angle (ASA) Triangle Congruence Postulate, $\triangle RTS \cong \triangle UTV$.

Answer:

$\triangle RTS \cong \triangle UTV$ is proven by the ASA Triangle Congruence Postulate.